Optimal. Leaf size=222 \[ -\frac {2166399 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{7700}-\frac {140289 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3850}-\frac {1341}{154} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {6547351 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3500}-\frac {722133 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3500} \]
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Rubi [A]
time = 0.06, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 159,
164, 114, 120} \begin {gather*} -\frac {722133 \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3500}-\frac {6547351 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3500}+\frac {(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}-\frac {56 (5 x+3)^{3/2} (3 x+2)^{5/2}}{11 \sqrt {1-2 x}}-\frac {1341}{154} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}-\frac {140289 \sqrt {1-2 x} (5 x+3)^{3/2} \sqrt {3 x+2}}{3850}-\frac {2166399 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{7700} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x} \left (\frac {93}{2}+75 x\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-6285-\frac {20115 x}{2}\right ) (2+3 x)^{3/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {1341}{154} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {\int \frac {\sqrt {2+3 x} \sqrt {3+5 x} \left (\frac {2664975}{4}+\frac {2104335 x}{2}\right )}{\sqrt {1-2 x}} \, dx}{1155}\\ &=-\frac {140289 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3850}-\frac {1341}{154} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {\int \frac {\left (-\frac {190065795}{4}-\frac {292463865 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{28875}\\ &=-\frac {2166399 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{7700}-\frac {140289 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3850}-\frac {1341}{154} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {\int \frac {\frac {12310799985}{8}+\frac {9722816235 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{259875}\\ &=-\frac {2166399 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{7700}-\frac {140289 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3850}-\frac {1341}{154} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {2166399 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{7000}+\frac {6547351 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3500}\\ &=-\frac {2166399 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{7700}-\frac {140289 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3850}-\frac {1341}{154} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {6547351 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3500}-\frac {722133 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3500}\\ \end {align*}
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Mathematica [A]
time = 9.21, size = 130, normalized size = 0.59 \begin {gather*} -\frac {10 \sqrt {2+3 x} \sqrt {3+5 x} \left (1041609-2751916 x+567906 x^2+198180 x^3+40500 x^4\right )+13094702 \sqrt {2-4 x} (-1+2 x) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-6595505 \sqrt {2-4 x} (-1+2 x) F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{21000 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 239, normalized size = 1.08
method | result | size |
default | \(-\frac {\left (12998394 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-26189404 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+6075000 x^{6}-6499197 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+13094702 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+37422000 x^{5}+125270100 x^{4}-292994460 x^{3}-332548330 x^{2}+32790750 x +62496540\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}}{21000 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\) | \(239\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-\frac {135 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{28}-\frac {1989 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{70}-\frac {265487 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2800}+\frac {2763367 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{9800 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {6547351 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{14700 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {-\frac {57575}{8} x^{2}-\frac {218785}{24} x -\frac {11515}{4}}{\sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {3773 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{192 \left (-\frac {1}{2}+x \right )^{2}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(298\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.25, size = 55, normalized size = 0.25 \begin {gather*} -\frac {{\left (40500 \, x^{4} + 198180 \, x^{3} + 567906 \, x^{2} - 2751916 \, x + 1041609\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2100 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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